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# complex.h
**`complex.h`** is what you include when you need the maths to work directly with complex numbers, rather than manually tracking real and imaginary parts in separate `double` variables. Added in C99, it gives you `_Complex` types, the imaginary unit `I`, and the full set of functions for magnitude, phase, conjugate, exponential, trig, and so on.

```c
#include <complex.h>

double complex z = 3.0 + 4.0 * I;   // I is sqrt(-1)

double mag   = cabs(z);       // magnitude: sqrt(3² + 4²) = 5.0
double phase = carg(z);       // phase angle: atan2(4, 3) ≈ 0.9273 rad
double complex c = conj(z);   // conjugate: 3 - 4i
```

The arithmetic operators work natively on `double complex` — addition, subtraction, multiplication, and division all just work. The full function set mirrors `<math.h>`: `csin`, `ccos`, `cexp`, `clog`, `cpow`, `csqrt`, `creal`, `cimag`. Float variants have an `f` suffix (`cabsf`) and long double variants have an `l` suffix.

Complex numbers come up in signal processing (FFT, filter design), electrical engineering (impedance), and quantum mechanics (wave functions). For heavy numerical work, libraries like FFTW provide the optimised implementations; `<complex.h>` is the language-level foundation they sit on top of.

## Practice

```c
// compile: gcc -o cplex cplex.c -lm
// run: ./cplex
// description: basic complex arithmetic; Euler's formula as a sanity check

#include <complex.h>
#include <math.h>
#include <stdio.h>

int main(void) {
    double complex z = 3.0 + 4.0 * I;

    printf("|z|     = %.1f\n",     cabs(z));            // 5.0
    printf("arg(z)  = %.4f rad\n", carg(z));            // 0.9273
    printf("conj(z) = %.1f%+.1fi\n", creal(conj(z)), cimag(conj(z)));

    // Euler's formula: e^(i*pi) should equal -1 + 0i
    double complex euler = cexp(I * M_PI);
    printf("e^(i*pi) = %.6f%+.6fi\n", creal(euler), cimag(euler));
    return 0;
}
```

The last line demonstrates Euler's formula: $e^{i\pi} \approx -1$. The imaginary part will be a very small number near zero but not exactly zero — floating-point rounding applies to complex arithmetic just as it does to real arithmetic. If you are seeing values like `-1.000000+0.000000i`, that tiny residue is hidden by the format width; try `%.20f` to see it.